I was trying to get an expression for the loss of intensity due to vignetting in a simple optical system, and got a fairly complex integral. I was wondering if there's an easier way, or any book that does this calculation. I guess not, because basic books as Hetch or Born & Wolf don't give the expression, but it doesn't hurt to ask.
The system is simply an extended light source at the focal plane of a converging lens, a second converging lens at a certain distance, and a detector at the focal plane of the second lens. Everything is circular, both lenses are ideal and have the same diameter, and the detector is big enough. I'm interested in an isotropic light source with a Gaussian profile (i.e., a fluorescence emission from a surface when illuminated by a focused laser), but you can assume a different angular or radial profile if that makes things simpler.
If this is not usually done by hand, but you use an optical design program to get the result for each case, please do tell me which do you use.
In short, the questions are:
- What is the an analytical expression for this?
- Is there any book that explicitly makes the calculation?
- If this is usually done with a specialized program, which one do you recommend?
Thanks in advance